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Lesson: Linear Motion with Definite Integrals

Sample Question Videos

Worksheet • 25 Questions • 3 Videos

Q1:

A particle is moving in a straight line such that its speed at time 𝑑 seconds is given by Given that its initial position π‘Ÿ = 1 6 0 m , find its position when 𝑑 = 3 s e c o n d s .

Q2:

A particle started moving in a straight line from the origin such that its acceleration at time 𝑑 seconds is given by Given that its initial velocity was 14 m/s, determine its velocity 𝑣 and its displacement 𝑠 when 𝑑 = 2 s e c o n d s .

  • A 𝑣 = 2 2 / m s , 𝑠 = 3 2 m
  • B 𝑣 = 8 / m s , 𝑠 = 3 2 m
  • C 𝑣 = 8 / m s , 𝑠 = 1 2 m
  • D 𝑣 = 2 2 / m s , 𝑠 = 1 2 m

Q3:

The diagram below shows the acceleration of a particle which was initially at rest. What was its velocity at 𝑑 = 7 s ?

Q4:

The acceleration-time graph of a particle which was initially at rest is shown below. What was its velocity at 𝑑 = 1 1 s ?

Q5:

A particle is moving in a straight line such that its velocity at time 𝑑 seconds is given by Given that its initial position π‘Ÿ = 1 3 0 m , find an expression for its position at time 𝑑 seconds.

  • A  1 4 𝑑 + 1 4 ( 4 𝑑 ) + 5 1 4  c o s m
  • B [ βˆ’ 4 ( 4 𝑑 ) + 1 3 ] c o s m
  • C [ βˆ’ 4 ( 4 𝑑 ) + 1 7 ] c o s m
  • D  1 4 ( 4 𝑑 ) + 5 1 4  c o s m

Q6:

A particle started moving in a straight line from point towards point . Its velocity after seconds is given by After 2 seconds, another particle started moving in a straight line from point towards point . This particle was accelerating at 0.9 m/s2. The two particles collided 6 seconds after the first particle started moving. Find the distance .

Q7:

A particle is moving in a straight line such that its velocity at time 𝑑 seconds is given by Given that when 𝑑 = πœ‹ 2 seconds its position from a fixed point is π‘Ÿ = 2 0 3 m , find an expression for π‘Ÿ in terms of 𝑑 .

  • A s i n ο€Ό 4 𝑑 πœ‹  + 2 0 3
  • B βˆ’ 1 6 πœ‹ ο€Ό 4 𝑑 πœ‹  + 2 0 3 2 s i n
  • C βˆ’ ο€Ό 4 𝑑 πœ‹  + 2 0 3 s i n
  • D πœ‹ 4 ο€Ό 4 𝑑 πœ‹  + 2 0 3 s i n

Q8:

A particle started moving in a straight line. Its acceleration at time 𝑑 seconds is given by Find the maximum velocity of the particle 𝑣 m a x and the distance π‘₯ it travelled before it attained this velocity.

  • A 𝑣 = 1 0 3 / m a x m s , π‘₯ = 2 5 1 2 m
  • B 𝑣 = 1 0 3 / m a x m s , π‘₯ = 1 0 3 m
  • C 𝑣 = 2 0 3 / m a x m s , π‘₯ = 1 0 3 m
  • D 𝑣 = 2 0 3 / m a x m s , π‘₯ = 2 5 1 2 m

Q9:

A particle started moving from rest along the π‘₯ -axis from a point at π‘₯ = 1 0 m . Its acceleration at time 𝑑 seconds is given by Express its velocity 𝑣 and its displacement π‘₯ after time 𝑑 seconds.

  • A 𝑣 = ο€Ή 5 𝑑 + 5 𝑑  / 2 m s , π‘₯ = ο€Ύ 5 𝑑 3 + 5 𝑑 2 + 1 0  3 2 m
  • B 𝑣 = 1 0 𝑑 / m s , π‘₯ = ( 1 0 𝑑 + 1 0 ) m
  • C 𝑣 = ο€Ή βˆ’ 5 𝑑 βˆ’ 5 𝑑  / 2 m s , π‘₯ = ο€Ύ βˆ’ 5 𝑑 3 βˆ’ 5 𝑑 2 βˆ’ 1 0  3 2 m
  • D 𝑣 = ο€Ή βˆ’ 5 𝑑 βˆ’ 5 𝑑  / 2 m s , π‘₯ = ο€Ύ βˆ’ 5 𝑑 3 βˆ’ 5 𝑑 2 + 1 0  3 2 m

Q10:

A particle, starting from rest, began moving in a straight line. Its acceleration π‘Ž , measured in metres per second, and the distance π‘₯ from its starting point, measured in metres, satisfy the following equation π‘Ž = π‘₯ 1 5 2 . Find the speed 𝑣 of the particle when π‘₯ = 1 1 m .

  • A 𝑣 = 1 1 √ 1 1 0 1 5 / m s
  • B 𝑣 = 1 2 1 4 5 / m s
  • C 𝑣 = 2 4 2 1 5 / m s
  • D 𝑣 = 1 1 √ 5 5 1 5 / m s

Q11:

A particle is moving in a straight line such that its acceleration at time 𝑑 seconds is given by Given that its initial velocity is 1 2 1 3 m/s and its initial position from a fixed point is 7 1 0 m, determine its position when 𝑑 = 2 πœ‹ seconds.

  • A 7 1 0 m
  • B βˆ’ 7 1 0 m
  • C βˆ’ 1 0 7 m
  • D 1 0 7 m

Q12:

The figure shows a velocity-time graph for a particle moving in a straight line. Find the magnitude of the displacement of the particle.

Q13:

A particle is moving in a straight line such that its velocity after 𝑑 seconds is given by Find the distance covered during the time interval between 𝑑 = 0 s and 𝑑 = 1 2 s .

Q14:

The acceleration of a particle moving in a straight line, at time 𝑑 seconds, is given by When 𝑑 > 1 3 , the particle moves with uniform velocity 𝑣 . Determine the velocity 𝑣 and the distance, 𝑑 , covered by the particle in the first 23 s of motion.

  • A 𝑣 = 2 5 3 . 5 / c m s , 𝑑 = 4 7 3 2 c m
  • B 𝑣 = 7 6 0 . 5 / c m s , 𝑑 = 9 8 0 2 c m
  • C 𝑣 = 2 5 3 . 5 / c m s , 𝑑 = 4 2 3 2 c m
  • D 𝑣 = βˆ’ 6 9 0 / c m s , 𝑑 = 4 2 3 2 c m
  • E 𝑣 = βˆ’ 6 9 0 / c m s , 𝑑 = 4 7 0 3 c m

Q15:

A body started moving along the π‘₯ -axis from the origin at an initial speed of 10 m/s. When it was 𝑠 meters away from the origin and moving at 𝑣 m/s, its acceleration was ( 4 5 𝑒 ) βˆ’ 𝑠 m/s2 in the direction of increasing π‘₯ . Determine 𝑠 when 𝑣 = 1 1 / m s .

  • A l n ο€Ό 3 0 2 3  m
  • B l n ο€Ό 2 0 2 3  m
  • C l n ο€Ό 2 3 3 0  m
  • D l n ο€Ό 2 3 2 0  m
  • E l n ο€Ό 6 2 3  m

Q16:

A body moves in a straight line. At time 𝑑 seconds, its acceleration is given by Given that the initial displacement of the body is 9 m, and when 𝑑 = 2 s , its velocity is 27 m/s, what is its displacement when 𝑑 = 3 s ?

Q17:

A particle is moving in a straight line such that its acceleration, π‘Ž metres per second squared, and displacement, π‘₯ metres, satisfy the equation π‘Ž = 2 6 𝑒 βˆ’ π‘₯ . Given that the particle's velocity was 12 m/s when its displacement was 0 m, find an expression for 𝑣 2 in terms of π‘₯ , and determine the speed 𝑣 m a x that the particle approaches as its displacement increases.

  • A 𝑣 = 1 9 6 βˆ’ 5 2 𝑒 2 βˆ’ π‘₯ , 𝑣 = 1 4 / m a x m s
  • B 𝑣 = 1 9 6 βˆ’ 2 6 𝑒 2 βˆ’ π‘₯ , 𝑣 = 1 4 / m a x m s
  • C 𝑣 = 1 7 0 βˆ’ 2 6 𝑒 2 βˆ’ π‘₯ , 𝑣 = 1 3 / m a x m s
  • D 𝑣 = 1 7 0 βˆ’ 5 2 𝑒 2 βˆ’ π‘₯ , 𝑣 = 1 3 / m a x m s

Q18:

A particle moves in a straight line. At time 𝑑 seconds, its velocity, in meters per second, is given by What is its displacement in the interval 0 ≀ 𝑑 ≀ πœ‹ 2 s e c o n d s ?

Q19:

A particle started moving in a straight line such that its acceleration, measured in metres per second, and its displacement from a fixed point, measured in metres, satisfy the following equation: π‘Ž = π‘₯ βˆ’ 6 . Given that its initial displacement from the fixed point was 9 metres and its initial velocity was 5 m/s, determine its velocity when π‘Ž = 0 .

Q20:

A particle moves along the π‘₯ axis. It is initially at rest at the origin. At time 𝑑 seconds, the particle’s acceleration is given by How long does it take for the particle to return to the origin?

  • A 7 2 1 9 s
  • BThe particle will never return to the origin.
  • C 3 6 1 9 s
  • D 4 8 1 9 s
  • E 2 4 1 9 s

Q21:

A particle started moving from a fixed point in a straight line such that its acceleration π‘Ž , measured in metres per second squared, and its position π‘₯ , measured in metres, satisfy the following equation: π‘Ž = 2 π‘₯ + 2 4 . Given that the initial velocity of the particle was 2 m/s, find an expression for 𝑣 2 in terms of π‘₯ .

  • A 𝑣 = 2 π‘₯ + 4 8 π‘₯ + 4 2 2
  • B 𝑣 = 2 π‘₯ + 4 8 π‘₯ + 2 2 2
  • C 𝑣 = π‘₯ + 2 4 π‘₯ + 2 2 2
  • D 𝑣 = π‘₯ + 2 4 π‘₯ + 4 2 2

Q22:

A particle starts from rest and moves in a straight line. At time 𝑑 seconds, its velocity is given by How long does it take for the particle to return to its starting point?

Q23:

The given acceleration-time graph represents the motion of a particle. Given that the particle moved in a straight line and was at rest at time 𝑑 = 0 s , find the velocity of the particle at 𝑑 = 1 2 s .

Q24:

A particle started moving on the π‘₯ -axis from the origin 𝑂 at an initial velocity 𝑣 0 . At time 𝑑 seconds, the particle’s acceleration is given by Given that after 6 seconds, the particle’s displacement was 75 m in the right from point 𝑂 , Determine 𝑣 0 . If the body passed by 𝑂 once again, determine its velocity 𝑣 at this moment.

  • A 𝑣 = 4 9 2 / 0 m s , 𝑣 = 0 / m s
  • B 𝑣 = βˆ’ 2 3 9 2 / 0 m s , 𝑣 = 1 4 7 2 / m s
  • C 𝑣 = 0 / 0 m s , 𝑣 = 4 9 2 / m s
  • D 𝑣 = βˆ’ 4 9 2 / 0 m s , 𝑣 = βˆ’ 2 3 9 2 / m s
  • E 𝑣 = βˆ’ 2 3 9 2 / 0 m s , 𝑣 = βˆ’ 4 9 2 / m s

Q25:

Two bodies, initially at rest at the same point, start to move in the same direction along the same straight line. At time 𝑑 seconds ( 𝑑 β‰₯ 0 ) , their velocities are given by 𝑣 = ( 2 3 𝑑 βˆ’ 5 8 8 ) / 1 c m s and 𝑣 = ο€Ή 3 𝑑 βˆ’ 6 1 𝑑  / 2 2 c m s . Determine the time taken for the two bodies to be 2 744 cm apart.

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